The Americans birdied the first three holes to race into a 3-up lead before McIlroy got one back by being the only player to make par on the sixth.
At the par 5 seventh, Mickelson holed a birdie putt from 15 feet before McDowell lipped out from five feet to restore a 3-up lead.
Bradley then continued his remarkable form on the greens by making a 25-footer on the next but McDowell birdied nine to keep the lead to three as they went into the back nine.
Holes were halved on 10, 11, 12 and 13 before McIlroy knocked in a birdie four on the 15th to reduce the deficit to two.
The world number one's long-range effort on the next just crept by the hole and so Europe's flagship pairing were dormie 2 heading to the picturesque par-three 17th.
McIlroy sent his tee shot pin high to within 10 feet, with fellow Northern Irishman getting his ball wet with a short effort.
McIlroy looked good for a birdie two but Mickelson plonked his shot within a few inches to guarantee a 2&1 triumph and extend the hosts' lead in the overall standings to 4-2.